The maximum value of the function defined by \( f(x)=\min \) \( \left(e^{x}, 2+e^{2}-x, 8\right)...

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The maximum value of the function defined by \( f(x)=\min \) \( \left(e^{x}, 2+e^{2}-x, 8\right) \) is \( \alpha \) then integral value of \( x \) satisfying the inequality \( \frac{x(x-[\alpha])}{x^{2}-[\alpha] x+12}0 \) is
Note: \( [k] \) denotes greatest integer function less than or equal to \( k \).
(a) 1
(b) 3
(c) 5
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